Symmetry, Integrability and Geometry: Methods and Applications
Vol. 1 (2005), Paper 012, 10 pages
Radiation Reaction, Renormalization
and Poincaré Symmetry
Yurij YAREMKO
Institute for Condensed Matter Physics of National Academy of Sciences of Ukraine,
1 Svientsitskii Str., Lviv, 79011 Ukraine
E-mail: yar@ph.icmp.lviv.ua
Received July 08, 2005, in final form October 23, 2005; Published online November 01, 2005
Original article is available at http://www.emis.de/journals/SIGMA/2005/Paper012/
Abstract. We consider the self-action problem in classical electrodynamics of a massive
point-like charge, as well as of a massless one. A consistent regularization procedure is proposed, which exploits the symmetry properties of the theory. The radiation reaction forces in
both 4D and 6D are derived. It is demonstrated that the Poincaré-invariant six-dimensional
electrodynamics of the massive charge is renormalizable theory. Unlike the massive case,
the rates of radiated energy-momentum tend to infinity whenever the source is accelerated.
The external electromagnetic fields, which do not change the velocity of the particle, admit only its presence within the interaction area. The effective equation of motion is the
equation for eigenvalues and eigenvectors of the electromagnetic tensor. The interference
part of energy-momentum radiated by two massive point charges arbitrarily moving in flat
spacetime is evaluated. It is shown that the sum of work done by Lorentz forces of charges
acting on one another exhausts the effect of combination of outgoing electromagnetic waves
generated by the charges.
Key words: classical electrodynamics; point-like charges; Poincaré invariance; conservation
laws; renormalization procedure
2000 Mathematics Subject Classification: 70S10; 78A40
1
Introduction
In classical electrodynamics particles interact with one another through the medium of a field,
which has its own uncountable infinite degrees of freedom. The dynamics of entire system is
governed by the action
I=
2 X
a=1
Z
Ia,part + ea
dτa Aµ żaµ
−
1
16π
Z
d4 yfµν f µν ,
(1)
which is invariant under ten infinitesimal transformations (space-time translations and rotations)
that constitute the Poincaré group. These symmetry properties imply the conservation laws,
i.e. those quantities that do not change with time. (In 4-dimensional Minkowski space Ia,part
is proportional to the worldline length, while in six dimensions the particle part of action (1)
involves the term which is proportional to the curvature of particle’s world line [1, 2].)
Since the field fa,µν := ∂µ Aa,ν − ∂ν Aa,µ generated by charge ea has a singularity on its world
line, demanding that the total action (1) be stationary under a variation δzaµ (τa ) of the world
line does not give sensible motion equations. To make sense of the retarded field’s action on the
particle, we should perform the regularization procedure. It can be accomplished via substitution
of Liénard–Wiechert solutions for field variables in ten conserved quantities corresponding to
2
Yu. Yaremko
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Figure 1. We start with standard action principle used in classical field theory. Variation with respect to
field variables (V.f.v.) yields wave equations. The Poincaré invariance assures us, via Noether’s theorem
(N.T.), of conservation laws. The solutions of wave equations with point-like sources are substituted for
field variables in these quantities, which do not change with time. Having performed renormalization
procedure we derive the effective equations of motion which involve particles’ characteristics only.
the symmetry of the action (1) under the Poincaré group (see Fig. 1). The conservation of the
energy-momentum carried by the electromagnetic field [3]
Z
ν
pem (τ ) =
dσµ T µν
Σ
and the angular momentum tensor of the electromagnetic field [3]
Z
µν
Mem (τ ) =
dσα (xµ T αν − xν T αµ )
Σ
corresponds to the symmetry of action (1) with respect to space-time translations and spacetime rotations, respectively. (By T̂ we denote the electromagnetic field’s stress-energy tensor;
dσµ is the outward-directed surface element on an arbitrary space-like hypersurface Σ.)
Noether quantity Gαem carried by electromagnetic field consists of terms of two quite different
types [4]: (i) bound, Gαbnd , which are permanently “attached” to the sources and carried along
with them; (ii) radiative, Gαrad , which detach themselves from the charges and lead independent
existence. Within regularization procedure the bound terms are coupled with energy-momentum
and angular momentum of “bare” sources, so that already renormalized characteristics Gαpart of
charged particles are proclaimed to be finite. Noether quantities, which are properly conserved,
become:
Gα = Gαpart + Gαrad .
In the present paper we apply the regularization scheme based on the balance equations Ġα = 0
to the following problems:
• self-action in four- and six-dimensional classical electrodynamics;
• interference in the radiation of two point-like sources in 4-dimensional Minkowski space;
• radiation reaction for a photon-like charge in response to the external electromagnetic
field.
2
On the regularization procedure in classical electrodynamics
We do not need assumptions concerning with the structure of singularity of the Maxwell energymomentum tensor density T̂ on a particle trajectory ζ : R → M 4 . Our consideration is founded
on the energy-momentum and angular momentum balance equations for a closed system of
Radiation Reaction, Renormalization and Poincaré Symmetry
3
accelerated charge e coupled with electromagnetic field. They constitute the following system
of differential equations [5, 6]:
2
ṗµpart = − e2 z̈ 2 ż µ
3
2
ż ∧ ppart = − e2 ż ∧ z̈.
3
(2)
(3)
(Symbol ∧ denotes the wedge product; we use an overdot on z to indicate differentiation with
respect to the evolution parameter τ .) The solution of equation (3) involves an arbitrary scalar
function, say m:
2
pµpart = m(τ )ż µ − e2 z̈ µ .
3
(4)
This expression explains how already renormalized particle 4-momentum depends on worldline
characteristics, i.e. on 4-velocity ż := u and 4-acceleration z̈ := a.
Since ż 2 = −1, the scalar product of particle 4-velocity on the first-order derivative of particle
4-momentum (2) is as follows:
2
(ṗpart · ż) = e2 z̈ 2 .
3
(5)
Since (ż · z̈) = 0, the scalar product of particle acceleration on the particle 4-momentum (4) is
given by
2
(ppart · z̈) = − e2 z̈ 2 .
3
(6)
Summing up (5) and (6), we obtain
d(ppart · ż)
= 0.
dτ
(7)
According to [7], the scalar product of the 4-momentum and the 4-velocity (ppart · ż) = m(τ ) is
the rest mass of the particle. Since (7) the function m is of constant value. Having substituted
the expression (4) with m = const into equation (2), we obtain the well-known Abraham 4-vector
of radiation reaction [8].
3
Interference in the radiation of two point-like sources
The Maxwell energy-momentum tensor density T̂ is quadratic in electromagnetic field strengths
fˆ = fˆ(1) + fˆ(2) where fˆ(a) denotes the retarded Liénard–Wiechert solution associated with the
a-th particle. Hence T̂ can be decomposed as follows [9]:
µν
µν
µν
T µν = T(1)
+ T(2)
+ Tint
.
By T̂(a) we mean the contribution due to the field of the a-th particle [3]
1
µν
µλ ν
κλ (a)
4πT(a)
= f(a)
f(a)λ − η µν f(a)
fκλ
4
while interference term
1
µν
µλ ν
µλ ν
κλ (2)
κλ (1)
4πTint
= f(1)
f(2) λ + f(2)
f(1) λ − η µν f(1)
fκλ + f(2)
fκλ
4
describes the combination of outgoing electromagnetic waves.
(8)
4
Yu. Yaremko
Aguirregabiria and Bel [9] prove the fundamental theorem that the “mixed” radiation rate
does not depend on the shape of the space-like surface, which is used to integrate the Maxwell
energy-momentum tensor density. To integrate the “mixed” radiation rate
Z
µν
ν
dσµ Tint
,
pint =
Σ
we use the hyperplane Σt = {y ∈ M 4 : y 0 = t} associated with an unmoving inertial observer.
The “laboratory” time t is a single common parameter defined along all the world lines ζa : R →
M 4 of the system. Since the delay in action ensures shifts in arguments in the electromagnetic
field strengths, a-th world line ζa is parametrized by “individual” retarded time ta .
The key to the problem is that the angle integration of the interference contribution (8)
results the sum of partial derivatives in time variables:
2π
Z
0ν
dϕTint
=
0
∂ 2 Gν0
∂Gν1
∂Gν2
+
+
.
∂t1 ∂t2
∂t1
∂t2
It allows us to calculate the interference part of energy-momentum carried by two-particle electromagnetic field. It consists of the bound terms and radiative terms [11]:
pνint
=
pνint,bnd
−
XZ
b6=a
t
−∞
µ
dta Fba
The bound terms are absorbed by particles’ 4-momenta within the renormalization procedure.
The radiative part of energy-momentum carried by “two-particle” field consists of the integrals of individual Larmor relativistic rates over corresponding world lines and the work done
by Lorentz forces of point-like charges acting on one another [10, 11]:
XZ t
Z
2 X
2 2 t
µ
µ
µ
P =
pa,part (t) + ea
dta (aa · aa )ua (ta ) −
dta Fba
.
3
−∞
−∞
µ
a=1
(9)
b6=a
Here pa,part denotes (already renormalized) four-momentum of the a-th charged particle. The
integral of Larmor relativistic rate describes the contributions T̂(a) due to the a-th individual
field, while the sum of work done by Lorentz force due to the b-th particle acting on the a-th
one expresses the joint contribution T̂int due to combination of fields.
Having differentiated equation (9), we arrive at the relativistic generalization of Newton’s
second law
2
µ
ṗµa,part = − e2a (aa · aa )uµa + Fba
,
3
where loss of energy due to radiation is taken into account.
4
Radiation reaction and renormalization in six-dimensional
classical electrodynamics
The Liénard–Wiechert potential in six dimensions depends on particle acceleration [1]; in mostly
plus signature
aµ (τ ) uµ (τ )
Aµ = e
+
(1
+
ra
)
.
k
r2
r3
Radiation Reaction, Renormalization and Poincaré Symmetry
5
Here r is the retarded distance [3] and ak = aα k α is the component of the particle acceleration
in the direction of k α := r−1 (y α − z α (τ )).
We assume that an intrinsic structure of singularity of T̂ in the immediate vicinity of particle’s world line ζ : R → M 6 is beyond the limits of classical theory. The radiative part of
electromagnetic field’s energy-momentum is as follows [12, 13]:
Z t
4
6
3 µ
µ
2
µ
µ
·
2 µ
dτ
prad = e
(10)
(ȧ · ȧ)u − (a · a)ȧ + a (a · a) + 2(a · a) u ,
5
35
7
−∞
where (a·a)· denotes the derivative of the square of particle acceleration with respect to evolution
parameter τ .
The radiative part of electromagnetic field’s angular momentum depends on all previous
motion of a source up to the instant of observation t [12, 13]:
Z t
µν
µ 2
ν
Mrad = e
dτ z µ Prad
− z ν Prad
−∞
Z t
4 µ ν
64
ν µ
µ ν
ν µ
(a ȧ − a ȧ ) + (a · a) (u a − u a ) .
+
dτ
(11)
5
35
−∞
Here Prad denotes the integrand of equation (10).
With (11) in mind we assume that already renormalized angular momentum tensor of the
particle has the form
µν
µ
ν
Mpart
= z µ pνpart − z ν pµpart + uµ πpart
− uν πpart
.
In [2] the extra momentum πpart is due to additional degrees of freedom associated with acceleration involved in Lagrangian function for rigid particle.
Scrupulous analysis of consistency of the energy-momentum balance equation and angular
momentum balance equation reveals that six-momentum of charged particle contains two (already renormalized) constants, m and µ:
3
8 β
64
β
β
β
β
2 4 β
·
β
ppart = mu + µ −ȧ + (a · a)u + e
ä − u (a · a) − (a · a)a .
(12)
2
5
5
35
Having substituted the right-hand side of (12) for the particle’s six-momentum in the energymomentum balance equation ṗpart + ṗrad = 0, we derive the Lorentz–Dirac equation of motion
of a charged particle under the influence of its own electromagnetic field. An external device
adds covariant external force Fext to the right-hand side of this expression [1]:
6
3
4 µ 2
µ
ṗµpart + e2
u ȧ − a2 ȧµ + aµ (a2 )· + 2a4 uµ = Fext
.
5
35
7
5
Radiation reaction and renormalization for a photon-like
charged particle
Our consideration is based on the Maxwell equations with point-like source [14, equation (14)]
2Aµ (x) = −4πjµ (x),
which governs the propagation of the electromagnetic field produced by a photon-like charge.
Current density is zero everywhere, except at the particle’s position where it is infinite
Z
jµ (x) = q dτ uµ (τ )δ[x − z(τ )]
6
Yu. Yaremko
and 2 := η αβ ∂α ∂β is the wave operator. Unlike the massive case, the photon-like charge, say q,
generates the far electromagnetic field
a ∧ k + ak u ∧ k
fˆ = q
,
r
(13)
which does not yield to divergent Coulomb-like self-energy. Hence the photon-like charge does
not possess an electromagnetic “cloud” permanently attached to it. The renormalization procedure is not necessary because the photon-like source is not “dressed”.
As a consequence, the Brink–Di Vecchia–Howe action term [15]:
Z
1
Ipart =
dτ e(τ )ż 2
(14)
2
is consistent with the field and the interaction terms (1). Variation of (14) with respect to
Lagrange multiplier e(τ ) 6= 0 yields the isotropy condition ż 2 = 0. The particle part (14) of the
total action (1) describes already renormalized massless charge.
The retarded distance involved in equation (13)
r = −ηαβ [xα − z α (s)]uβ (s)
(15)
contains particle’s characteristics (position, z α , and 4-velocity, uβ ) taken at the retarded time, s,
being the (causal) root of algebraic equation
ηαβ [xα − z α (s)][xβ − z β (s)] = 0.
Because of the isotropy condition, the retarded distance (15) vanishes on the ray in the direction
of particle’s 4-velocity taken at the instant of emission z(s) ∈ γ. Unlike the massive case, the
field (13) diverges at all the points of this ray with vertex at the point of emission.
Volume integration of the radiative energy-momentum tensor density
4πT αβ =
q2 2 α β
a k k
r2
over a hyperplane Σt = {x ∈ M 4 : x0 = t} gives the amount of radiated energy-momentum at
fixed instant t. We introduce the set of curvilinear coordinates for flat space-time M 4 involving
the observation time t and the retarded time s:
0
xα = z α (s) + (t − s)Ωα α0 nα .
The null vector n := (1, n) has the components (1, cos ϕ sin ϑ, sin ϕ sin ϑ, cos ϑ); ϑ and ϕ are two
polar angles which distinguish the points on the spherical wave front
(
)
X
0
2
i
i
2 0
S(z(s), t − s) = x ∈ M 4 : (x − s) =
(x − z (s)) , x = t ,
i
which is the intersection of the future light cones with vertex at point z(s) ∈ γ and hyperplane Σt .
Matrix space-time components are Ω0µ = Ωµ0 = δµ0 ; its space components Ωij constitute
the orthogonal matrix which rotates space axes of the laboratory Lorentz frame until new zaxis is directed along three-vector v. (Particle’s 4-velocity has the form (1, v i ), |v| = 1, if
parameterization of the world line γ is provided by a disjoint union of hyperplanes Σt .) In terms
of curvilinear coordinates (t, s, ϑ, ϕ) the retarded distance (15) is as follows:
r = (t − s)(1 − cos ϑ).
Radiation Reaction, Renormalization and Poincaré Symmetry
7
Figure 2. The bold circle pictures the trajectory of a photon-like charge. The others are spherical wave
fronts viewed in the observation hyperplane Σt = {x ∈ M 4 : x0 = t}. The circling photon-like charge
radiates infinite rates of energy-momentum and angular momentum in the direction of its velocity v at
the instant of emission. The energy-momentum carried by electromagnetic field of accelerated charge
tends to infinity on the spiral curve.
Angle integration reveals that the Larmor term tends to infinity whenever the photon-like
charge is accelerated:
Z t
q2
1
1
dτ a2 (τ ),
p0em =
− + lim
2
8 ϑ→0 2(1 − cos ϑ)2
−∞
Z t
2
q
3
1
1
i
pem =
− lim
−
dτ a2 (τ )v i (τ ).
2 8 ϑ→0 1 − cos ϑ 2(1 − cos ϑ)2
−∞
The situation is pictured in Fig. 2. Since the emitted radiation detaches itself from the charge
and leads an independent existence, it cannot be absorbed within a renormalization procedure.
To change the velocity of the massless charge the energy is necessary which is too large to
be observed, while non-accelerated photon-like charge does not generate the electromagnetic
field. The evolution of the particle beyond an interaction area is determined by the Brink–
Di Vecchia–Howe Lagrangian (14). The particle’s 4-momentum pµpart = e(τ )ż µ does not change
with time: ė(τ )ż µ = 0. Since ż µ 6= 0, the Lagrange multiplier e does not depend on τ . We deal
with a photon-like particle moving in the ż-direction with frequency ω0 = e0 ż 0 , such that its
energy-momentum 4-vector can be written pµpart = (ω0 , ω0 k), |k| = 1.
When considering the system under the influence of an external device, the change in particle’s
4-momentum is balanced by the external force:
ėv µ = qF µ ν v ν .
(16)
This effective equation of motion is supplemented with the condition of absence of radiative
damping. In other words, the external device admits a massless charge if and only if the
components of null vector of 4-velocity do not change with time despite the influence of the
external field.
The expression (16) is obtained in [16] where the model of magnetosphere of a rapidly rotating
neutron star (pulsar) is elaborated. (The gas of ultrarelativistic electrons and positrons moving
in a very strong electromagnetic field of the pulsar is meant.) If the gradient of star’s potential
8
Yu. Yaremko
is much larger than the particle’s rest energy me c2 , the strong radiation damping suppresses
the particle gyration. It is shown [16] that the particle velocity is directed along one of the
eigenvectors of (external) electromagnetic tensor F̂ if me → 0 in the Lorentz–Dirac equation.
Equation (16) on eigenvalues and eigenvectors of the electromagnetic tensor governs the motion
of charges in zero-mass approximation. This conclusion is in contradiction with that of Ref. [14],
where the radiation back reaction is finite and the 5-th order differential equation determines
the evolution of photon-like charge. The reason is that the Dirac regularization approach to
the radiation back reaction (one-half retarded field minus one-half advanced one), employed by
Kazinski and Sharapov, is not valid in the case of the photon-like charged particle and its field.
Indeed, the advanced field is free from the ray singularity, and the Dirac combination does not
yield a finite part of the self force.
5.1
Conformal invariance
Action integral (1) with Ipart in form of (14) is conformally invariant. According to [17], conformal group C(1, 3) consists of Poincaré transformations (time and space translations, space and
mixed space-time rotations), dilatations
x0µ = eθ xµ
(17)
and conformal transformations
x0µ =
xµ − bµ (x · x)
,
D
D = 1 − 2(x · b) + (x · x)(b · b).
(18)
(The scalar θ and 4-vector b are group parameters.)
The components of electromagnetic field are transformed as follows:
0
Fαβ = e2θ Fαβ
,
0
Fαβ = Fµν
Ωµ α Ων β
where matrix
Ωµ α :=
∂x0µ
= D−1 λµ β (x00 )λβ α (x),
∂xα
x00 =
x
− b,
(x · x)
λβ α (x) = δ β α −
2xβ xα
(x · x)
satisfies the condition
ηµν Ωµ α Ων β = D−2 ηαβ .
Since
0
ż µ = eθ ż µ
0
ż µ = Ωµ α ż α
the Lagrange multiplier e(τ ) involved in the Brink–Di Vecchia–Howe action term (14) transforms
as
e(τ ) = e2θ e0 (τ ),
e(τ ) = D−2 e0 (τ ).
Direct calculation shows that the effective equation of motion (16) is invariant with respect to
dilatation (17) and conformal transformation (18).
It is worth noting that the conformal invariance yields conservation laws, which are functions
of energy-momentum and angular momentum conserved quantities.
Conformal invariance of our particle plus field system reinforces our conviction that the backreaction force vanishes. Indeed, the appropriate renormalization procedure should preserve this
symmetry property while the the Brink–Di Vecchia–Howe action term (14) does not contain
a parameter to be renormalized. Therefore, the photon-like charge must not radiate.
Radiation Reaction, Renormalization and Poincaré Symmetry
6
9
Conclusions
The law of conservation of the total four-momentum of a composite (particle plus field) system
provides the foundation for Dirac’s derivation [8] of the radiation-reaction force. It involves the
Taylor expansion of a finite sized charged sphere, in which the first two terms lead to the electromagnetic self-energy, and the Abraham radiation reaction four-vector, respectively. Inevitable
infinities arising in six-dimensional electrodynamics are stronger than in four dimensions. In [1]
Kosyakov calculates the flux of energy-momentum and derives the radiation-reaction force in
six-dimensions by adding of appropriate Schott term.
In the present paper we consider also the conserved quantities corresponding to the invariance of the theory under proper homogeneous Lorentz transformations. They gives additional
information which allows us to avoid the problems concerning with an intrinsic structure of
a singularity. In such a way we reformulate the problem of renormalizability within the problem
of Poincaré invariance of a closed particle plus field system. The conservation laws are an immovable fulcrum about which tips the balance of truth regarding renormalization and radiation
reaction.
The renormalization scheme is applied to the problem of combination of outgoing electromagnetic waves generated by two point-like sources [10, 11]. The evaluation of the interference
part of energy-momentum radiated by charges is not a trivial matter, since delay in action ensures shifts in arguments in the electromagnetic field strengths. Rigorous calculations show that
the interference rate of radiation which escapes to infinity is equal to the rate of work done by
Lorentz forces of charges acting on one another. The bound terms arise, which can be interpreted as a usual deformation of the bound electromagnetic “clouds” of charges due to mutual
interaction. They are absorbed within the renormalization procedure, as well as the inevitable
infinities arising in a one-particle problem.
Inspection of the energy-momentum and angular momentum carried by the electromagnetic field of a photon-like charge reveals the reason why it is not yet detected (if it exists).
Noninteracting massless charges do manifest themselves in no way. Any external electromagnetic
field (including that generated by a detecting device) will attempt to change the velocity of the
charged particle. Whenever the effort will be successful, the radiation reaction will increase
extremely. In general, this circumstance forbids the presence of the photon-like charges within
the interaction area.
To survive, photon-like charges need an extremely strong field of specific configuration, as
that of the rotating neutron star (pulsar). In [16] the model of the pulsar magnetosphere is
elaborated. It involves the so-called dynamical phase, which consists of the massless charged
particles moving with speed of light along real eigenvalues of the electromagnetic field tensor of
the star.
Acknowledgements
The author would like to thank Professors B.P. Kosyakov, V. Tretyak, Dr. A. Duviryak, and
Professor Yu.A. Rylov for helpful discussions and critical comments.
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